On the limit of multiple packings

Abstract

For a collection

of subsets of a setS, letn _k be the maximal number of sets from

such that no point ofS belongs to more thank of them. Then the limitδ of the sequencen _k /k is related to the valuev of the game in which the minimizer selects a setα in

, the maximizer a pointx inS, and the payoff isϰ _α (x), byδ≤v ⁻¹≤inf _v δ _v . Herev is any finite subset ofS and δ _v the corresponding limit. Analogous results hold for covering.

The bounds obtained in this way forn _k and in particularn ₁ reflect the influence of the boundary ofS when, for instance,S is a convex body to be packed with congruent copies of another body. In some cases, the bounds are sharp.